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Lasry-Lions Envelopes and Nonconvex Optimization: A Homotopy Approach

Miguel Simões, Andreas Themelis, Panagiotis Patrinos

TL;DR

This work addresses nonsmooth nonconvex composite minimization by introducing Lasry-Lions double envelopes h^{λ,μ} as smooth surrogates that interpolate toward the convex hull of h. It develops a homotopy strategy φ_{λ,μ}=g+h^{λ,μ} and proves epi-convergence and convergence of minimizers as λ→0 with λ−μ→0, enabling a sequence of easier subproblems solved by smooth optimizers (e.g., L-BFGS). The authors provide theoretical properties of h^{λ,μ}, including Lipschitz-gradient behavior, and demonstrate practical performance on signal decoding and spectral unmixing, where the method is competitive with or superior to ADMM-based relaxations and offers flexible sparsity control. The approach broadens the toolkit for large-scale nonsmooth nonconvex optimization in imaging and communications, yielding a scalable, smooth surrogate framework with potential broad impact. The results suggest that Lasry-Lions envelopes can yield useful approximations and convergent homotopies for challenging inverse problems with nonconvex regularizers.

Abstract

In large-scale optimization, the presence of nonsmooth and nonconvex terms in a given problem typically makes it hard to solve. A popular approach to address nonsmooth terms in convex optimization is to approximate them with their respective Moreau envelopes. In this work, we study the use of Lasry-Lions double envelopes to approximate nonsmooth terms that are also not convex. These envelopes are an extension of the Moreau ones but exhibit an additional smoothness property that makes them amenable to fast optimization algorithms. Lasry-Lions envelopes can also be seen as an "intermediate" between a given function and its convex envelope, and we make use of this property to develop a method that builds a sequence of approximate subproblems that are easier to solve than the original problem. We discuss convergence properties of this method when used to address composite minimization problems; additionally, based on a number of experiments, we discuss settings where it may be more useful than classical alternatives in two domains: signal decoding and spectral unmixing.

Lasry-Lions Envelopes and Nonconvex Optimization: A Homotopy Approach

TL;DR

This work addresses nonsmooth nonconvex composite minimization by introducing Lasry-Lions double envelopes h^{λ,μ} as smooth surrogates that interpolate toward the convex hull of h. It develops a homotopy strategy φ_{λ,μ}=g+h^{λ,μ} and proves epi-convergence and convergence of minimizers as λ→0 with λ−μ→0, enabling a sequence of easier subproblems solved by smooth optimizers (e.g., L-BFGS). The authors provide theoretical properties of h^{λ,μ}, including Lipschitz-gradient behavior, and demonstrate practical performance on signal decoding and spectral unmixing, where the method is competitive with or superior to ADMM-based relaxations and offers flexible sparsity control. The approach broadens the toolkit for large-scale nonsmooth nonconvex optimization in imaging and communications, yielding a scalable, smooth surrogate framework with potential broad impact. The results suggest that Lasry-Lions envelopes can yield useful approximations and convergent homotopies for challenging inverse problems with nonconvex regularizers.

Abstract

In large-scale optimization, the presence of nonsmooth and nonconvex terms in a given problem typically makes it hard to solve. A popular approach to address nonsmooth terms in convex optimization is to approximate them with their respective Moreau envelopes. In this work, we study the use of Lasry-Lions double envelopes to approximate nonsmooth terms that are also not convex. These envelopes are an extension of the Moreau ones but exhibit an additional smoothness property that makes them amenable to fast optimization algorithms. Lasry-Lions envelopes can also be seen as an "intermediate" between a given function and its convex envelope, and we make use of this property to develop a method that builds a sequence of approximate subproblems that are easier to solve than the original problem. We discuss convergence properties of this method when used to address composite minimization problems; additionally, based on a number of experiments, we discuss settings where it may be more useful than classical alternatives in two domains: signal decoding and spectral unmixing.

Paper Structure

This paper contains 7 sections, 13 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Lasry--Lions double envelope
  3. A homotopy approach to nonconvex minimization
  4. Numerical experiments
  5. Signal decoding
  6. Spectral unmixing
  7. Conclusions

Figures (2)

  • Figure 1: Approximations with Lasry--Lions envelopes $h^{\lambda,\mu}$ (blue), proximal hulls $h^{\lambda,\lambda}$ (brown), and Moreau envelopes $h^\lambda$ (green) of the nonconvex, nonsmooth, extended-real-valued function $h=\mathop{\mathrm{\delta}}\nolimits_{\{\pm1\}}$ (black).
  • Figure 2: Average RMSE, sensitivity, and specificity.

Theorems & Definitions (5)

  • proof : Detail
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